Vibronic Exciton–Phonon States in Stack-Engineered van der Waals Heterojunction Photodiodes

Stack engineering, an atomic-scale metamaterial strategy, enables the design of optical and electronic properties in van der Waals heterostructure devices. Here we reveal the optoelectronic effects of stacking-induced strong coupling between atomic motion and interlayer excitons in WSe2/MoSe2 heterojunction photodiodes. To do so, we introduce the photocurrent spectroscopy of a stack-engineered photodiode as a sensitive technique for probing interlayer excitons, enabling access to vibronic states typically found only in molecule-like systems. The vibronic states in our stack are manifest as a palisade of pronounced periodic sidebands in the photocurrent spectrum in frequency windows close to the interlayer exciton resonances and can be shifted “on demand” through the application of a perpendicular electric field via a source-drain bias voltage. The observation of multiple well-resolved sidebands as well as their ability to be shifted by applied voltages vividly demonstrates the emergence of interlayer exciton vibronic structure in a stack-engineered optoelectronic device.


Section S1. WSe 2 /MoSe 2 Device Fabrication
In this work, we present intensive spectroscopic, electronic, and optoelectronic characterization of precisely fabricated heterostructure devices composed of bilayer WSe 2 and monolayer MoSe 2 . Based on our previous temperature dependent electronic transport study on the same material system 17,51 , this unique combination of materials was chosen due to its indirect band gap, unique phonon band structure, and long-lived interlayer excitons expected at infrared wavelengths.
The heterostructure devices are assembled using a highly customized, temperaturecontrolled transfer microscope that ensures the interface between the two layers has no intentional contact to polymer films. The first step of the fabrication is to make the back-gate assembly as shown in Fig. S1b. The back gate consists of a layer of graphite (2 nm thick) as the conductor and a hexagonal boron nitride (h-BN) layer (12 nm) on top as the gate dielectric. The graphite layer is picked up using the h-BN and mounted on the prefabricated gate contacts. Fig. S1a shows the device geometry with all components in two different perspectives. Fig. S1b shows optical images of the three stages of the fabrication process from left to right. The left image in Fig. S1b shows the first step of the fabrication which is the gate assembly on the SiO 2 substrate. The next step is to write source-drain contacts on top of the bottom h-BN layer, these layers are labeled as MoSe 2 contact and WSe 2 contact shown in Fig. S1b middle. These contacts provide electrical access to each layer of the stack. For device 1 the MoSe 2 /WSe 2 heterostructure was fabricated using mechanical exfoliation of WSe 2 and MoSe 2 flakes from bulk crystal (2D semiconductors) onto Si wafers coated with 290-nm-thick SiO 2 . The silicon substrate is implemented as a back gate to tune the carrier density at the interface. Electrical access to the two layers of MoSe 2 and WSe 2 is achieved through independent electrical contacts to each material. The details of the fabrication process can be found in our previous study 17 .
To minimize interfacial contamination, the two layers of the stack (WSe 2 and MoSe 2 ) are picked up via the top h-BN layer one after another in a two-step transferring process and mounted on the source drain contacts on the prefabricated gate. To get rid of possible impurities and destructive bubbles formed in between the layers of the heterostructure the device is finally annealed at 250 °C under Ar gas for one hour. The final step is to mount the device into a chip carrier within the measurement setup. The chip carrier is a patterned substrate of sapphire with gold for electrical access to the device. The electronic contacts on the SiO 2 substrate are wire bonded into the conductive pads of the chip carrier shown in Fig. S1 (left). The device is then soldered to the electronic pads of the cryostat and silver painted for further electronic experiments shown in Fig. S1c (right). (middle) for electrical access to the device. A stack of h-BN/WSe 2 /MoSe 2 is then laminated onto these prefabricated electrical contacts (right). c, Photographs of the devices wire-bonded into chip carriers. Device 2 (left) was contacted using a probe station cryostat (see Section S2.2). Device 1 (right) was contacted by soldering chip carrier to electrical pins of the cryostat (see section S2.1).

Section S2.1. The Supercontinuum Scanning Photocurrent Spectroscopy Microscope
The supercontinuum scanning photocurrent spectroscopy microscope (SSPSM) is a powerful tool designed to explore quantum optoelectronic properties of atomic scale devices by integrating optics and electronics into a combined quantum transport microscope. The SSPSM is equipped with a helium flow optical cryostat that enables both electrical access and optical illumination to nanoscale devices through electrical feedthroughs and a transparent window. This cryostat provides a stable and controllable low-pressure environment via a turbo pump system and can be cooled down to temperatures of T = 4 K.
The SSPSM microscope employs a supercontinuum Fianuim white laser as the light source along with Princeton Instrument monochromator for spatially and spectrally resolved supercontinuum excitations. The light source we use for this microscope is a Fianium WhiteLaseTM SC400-8; a picosecond pulsed laser pump (repetition rate = 20 MHz, pulse width = 6 ps) is coupled into a micro-structured optical fiber (the non-linear medium) of 1.5-meter length.
At the output of the fiber, the collimated white beam of light has an average power of 8 W and a power spectral density of 4 mW/nm that ranges from 400-2200 nm. To probe the optoelectronic response of the samples we use scanning beam photoexcitation, as shown in Fig. S2.1 (GVS, L7 and L8) to generate a collimated beam of light on the back aperture of an objective. As the angle of the collimated beam is shifted the resulting diffraction limited beam spot that is scanned over the nanoscale device. The resulting current is amplified using a sub-100 fA resolution pre-amplifier over long averaging times (10-1000 ms) at low optical power. The reflected light from the sample transmits through the BS and reflects to a lens (L9) and gets collected by a photodiode detector (PD To achieve the highest performance of this microscope over the broad spectral range (400-2200 nm), optical components must match the wavelength range that we choose to conduct experiments. There are three different beam splitter coatings to cover the whole wavelength range.
Thorlabs non-polarizing achromatic beam splitter cubes can provide a uniform reflection in these three ranges: BS028 for Vis (400-700 nm), BS029 for NIR (700-1100 nm), and BS030 for IR (1000-1600 nm). The beam splitters used for this microscope reflect 90% of the light to the sample; the 10% of the light that transmits through the BS can get detected with a power sensor (PS) coupled to a power meter (PM). The power sensor response function should also match the wavelength range of the microscope; using two power sensors (Thorlabs: S130C, and S132C) we can accurately detect the optical power in the ranges of 400-1100 nm and 700-1800 nm respectively in the power range of 500 pW-500 mW.
Data, in the form of two-dimensional maps of photocurrent and reflection, is acquired and displayed using python based custom control software 52 . Using the SSPSM microscope, we can perform spatial imaging as a function of wavelength by fixing the spectrum at a specific wavelength and conducting 2D spatial photocurrent mappings over the surface of the sample, while changing other parameters such as V SD , V G , or temperature in order to make 3D or 4D data cubes. We can also perform spectral imaging with a narrow resolution of 1 nm by fixing the laser beam at a specific spot on the device (e.g., the heterostructure as identified by correlating scanned reflection images to know device geometries) while tuning for example, the excitation wavelength.
Additional parameters such as V SD , V G , or temperature are then tuned to form a multi-dimensional data set.

Section S2.2. Multi-Parameter Dynamic Photoresponse Microscopy Measurement Details
Data intensive imaging of the sample phase space was performed using the Multi-Parameter Dynamic Photoresponse Microscopy (MPDPM) technique described previously by Arp & Gabor. Below we give a basic description of the most relevant aspects of the optics to this experiment, but more details on the hardware and methodology can be found in reference 52 . Data shown in the main figures is condensed from the MPDPM data analysis process, while the full imaging data sets can be found in section S4.3.
The core of the optical system is a Coherent MIRA 900 OPO ultrafast pulsed laser which generates 150 fs pulses with controllable wavelength from 1150 nm to 1550 nm at a 75 MHz repetition rate. The output of the laser is attenuated to control the laser power and fed into a scanning beam system similar to GVS, L7 and L8 in Fig. S2.1. After the second lens (L8 in Fig.   S2.1) the MPDPM optics differ significantly from the SSPSM setup in order to get a stronger optical probe. In MPDPM a strong and highly local optical probe is required for best results, therefore the output of the scanning optics is directed into a GRIN (Gradient Index of Refraction) lens, which performs a similar function to the objective lens but with dramatically lower pulse dispersion. Combined with the fact that the OPO laser output has significantly higher power than the output of the SSPSM monochromator (order of mW at the sample) the pulse fluence is several orders of magnitude larger in the MPDPM setup than the SSPSM.
Photocurrent was measured using a lock-in amplifier. The laser beam was modulated by an optical chopper and the signal was passed through a pre-amplifier then locked-in to the chopper frequency. This was necessary to remove the substantial dark current that occurs due to the sample diode characteristics (see Fig. 1f). The data was then acquired through a National Instruments PCIe-6323 Data Acquisition Card. In addition to the photocurrent signal, the reflected light from the sample was continuously acquired along with the laser power.
The imaging data set shown in Fig S4.3, along with several similar image sets, was acquired using the "datacube" method from MPDPM, i.e. automatically taking a series of 2D scanning images as a function of one variable (with all other variables held constant) to form a threedimensional data set, and then taking a series of these datacubes incremented over variable to get a 4-dimensional data set. Each set involved hundreds to thousands of images measured fully automatically over the course of several hours to a few days. For these longer measurements the main concern is laser stability, as any variation in the optics will cause the image location to drift.
To counter this, careful attention was paid to the alignment of the optics to avoid any geometrical aberration and the laser was run continuously for several days prior to any long imaging measurement to stabilize its performance. When the system is stable the main source of drift is thermal expansion of the various optical components; therefore, temperature and airflow of the lab is carefully maintained during measurements.
Even with the extra considerations for stability there will inevitably be some drift in the images, which must be corrected for in the data. There are multiple options but given that the laser power was monitored and shown to be constant throughout the measurement and the drift is small, the best method was to re-orient the data based on the reflection image. To find the correction for a given reflection image we use an algorithm that shifts the image by an integer number of pixels and calculates the difference between that image and the first image in the data set. By brute force the algorithm determines the shift that produces the minimum difference and uses that as the correction. This allows for correction that is accurate to the pixel resolution. For computational simplicity (and because the drift within datacubes was sub-pixel) we perform this procedure by comparing the first image in each 3D datacube to determine the shift for the whole cube.  and then the last image corrected to be spatially consistent with the first image (bottom). b, the required drift correction as a function of time over the measurement.

S3.1 Evidence for Phononic Origin of the Sidebands
In the WSe 2 -MoSe 2 heterojunction devices, we establish that photocurrent peaks arise from absorption resonances due to photoexcited interlayer excitons with dipole moments that are constrained to point out-of-plane. For both of these interlayer excitons, the observed energy spacing = 30 meV is similar as confirmed both by Lorentzian fitting (see Fig. 2 in the main Δ text, according to convention in the PL community) and by the Fourier transformed data. In contrast, similar periodic sideband behavior is not observed in the photocurrent due to intralayer excitons (See Supplementary Figure S4.1 and surrounding discussion below).
As a key point, the energy spacing does not seem to be sensitive to the distinct microscopic structure of the two types of interlayer excitons ( vs. ), and it is therefore highly unlikely → → that such periodic structure arises from random defect states below each of the exciton absorption (owing to the strong coincidence of energies within the electronic structure of two distinct interlayer excitons). The similarity in both excitons' spectral fingerprints indicates that periodic photocurrent peaks arise from a similar origin. To observe such sidebands due to potential modulation would require very careful fine tuning of the potential. We note that the side bands occur for a broad range of gate voltages, making fine-tuning effects unlikely.

S3.2 Phonon calculation details
In this Section, we investigate numerically the vibrational modes in the heterostructure in an effort to identify the candidate modes that can be involved in the vibronic-like photoresponse.
We note that this numerical study treats the electronic system in its insulating ground state (without excitons present) in order to identify phonon modes whose vibrational energies are consistent with the observed vibronic-like behavior. Vibrational coupling to the interlayer excitons at such large vibrational energies (~30 meV) suggests that the relevant vibrational modes are optical phonon modes with flat band structure on the scale of the thermal energy . Using detailed phonon dispersion calculations (described below) we identify such phonon modes (Einstein modes) within the MoSe 2 /WSe 2 heterostructure and explore the detailed properties of these modes. We find that the relevant modes exhibit flat bands with excitation energies near 30 meV.
The phonon dispersion of the heterostructure MoSe 2 /WSe 2 with spin-orbit coupling (SOC) was calculated using the Vienna ab initio simulation package (VASP) 53-55 in the projected-augmented-wave method 56 . The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) parametrization 57-59 was used for the exchange correlation energy. The Van der Waals interactions between the layers are accounted by using the DFT-D2 Grimme's method 60 . The kinetic energy cutoff for our calculation is set as 500 eV.
For all structural relaxations, the convergence tolerance on the Hellmann-Feynman forces is less than 10 -7 eV/Å. An 8×8×1 Γ-centered Monkhorst-Pack k-point mesh is used for 2D films.
The vacuum layer added to the system is nearly 20 Å. Under these settings, the lattice constant of MoSe 2 /WSe 2 is 3.3259 Å. The lattice mismatch between MoSe 2 and WSe 2 is less than 1%. Both of the two materials contain heavy elements; therefore, SOC is included for all calculations. The phonon dispersion is calculated for a 4×4×1 supercell using Phonopy 61 .

S3.3 Phonon dispersion relation and density of states
We first examine the phonon modes in the region near 30 meV. As previously reported, there are a variety of Raman active and Raman inactive phonon modes near that energy 62 . To   By comparing to our data, we identify candidate phonon modes that may couple strongly to an out-of-plane dipole. In Fig. 2

S4.1. Photocurrent Spectroscopy of Intralayer Excitons
In this section we compare the photocurrent response of intralayer excitons to that of interlayer excitons within the vdW heterojunction. We show that the intralayer exciton photocurrent exhibits several key differences when compared to the interlayer photocurrent: First, photocurrent due to intralayer excitons does not exhibit regularly spaced peaks like those shown for interlayer excitons as in Fig. 2. Second, photocurrent due to intralayer excitons does not require the device to be tuned to charge neutrality and exhibits gate voltage dependence for positively and negatively charged trion species, consistent with ordinary behavior. We observe features that originate in both the MoSe 2 and WSe 2 , with photocurrent excitation energies that match the relevant photoluminescence energies, consistent with ordinary photoresponse of intralayer excitons.
With high spectral resolution we tune the excitation energy close to the heterostructure peaks observed in the photoluminescence (Fig. 1b) and measure interlayer photocurrent. Fig. S4.1a shows photocurrent as a function of photon excitation energy from 1.55 eV to 1.37 eV (wavelength range between 800-900 nm), and V G between -20 to 20 V. We observed the appearance of three peaks at different photon energies that shift with gate voltage from V G = -20 V to 20 V. Notably, no sidebands are observed in the photoresponse of these peaks. range with a small slope, consistent with ordinary trion behavior. The photocurrent peak observed at 1.443 eV is attributed to the positive exciton (trion) X W + originating from WSe 2 . This peak again redshifts gradually as a function of gate voltage and corresponds closely to the PL peak position observed for WSe 2 . We note that none of these peaks are observed exclusively at charge neutrality, instead exhibiting gate voltage dependence consistent with ordinary behavior.

S4.2 Low Temperature Photocurrent Spectroscopy in the Vicinity of Charge Neutrality
The low temperature photocurrent spectroscopy data shown in Fig. 2 was measured using the SSPSM setup described in section S2.1 on device 1 in an optical cryostat cooled with liquid helium to approximately 20 Kelvin. The spectral resolution of E PH is limited by a tradeoff in the SSPSM monochromator where increasing the spectral resolution decreases the pulse power (or equivalently, the signal to noise).
In the low temperature measurements, the dominant current signal is the dark current of the device with respect to V G . Fig. S4.2a shows the raw current for the measurement shown in Fig.   2e (the Fig. 2a measurement is similar). We see that the raw signal is an order of magnitude larger than the photocurrent scale shown in Fig. 2e due to a large dark current. Fig. S4.2b shows that the photocurrent, i.e. the current varying with E PH, is a small modulation on top of the dark current.
There are several ways to process and remove the dark current, we use the simplest method, which is also advantageous for analyzing the photocurrent oscillations, that is we base our analysis on the derivatives with respect to E PH , where the constant dark current drops out. For the purposes of displaying the data in Fig. 2a,e (and Fig. S4.3a,e) we set the minimum of the color scale at some threshold (shown as the grey dashed line in Fig. S4.2b) that displays the photocurrent in the most active region of V G . This is purely visual, to give contrast to the photocurrent oscillations, and does not affect the subsequent analysis.   We now briefly discuss the electrostatic conditions of the device over the gate voltage range studied. In Figure 2, the voltage range shown is within the broad vicinity of charge neutrality, while the charge neutral condition occurs specifically within the narrower region between V G = -9 V and V G = 3 V. Fig. S4.3 shows that the sidebands are pronounced across the entire charge neutrality window. There is no marked change of the sidebands as V G is tuned across the charge neutrality window.
From previous work on vdW p-n heterojunctions (see for example, refs. 23-26), photocurrent in reverse bias -and dark current in forward bias -are expected to be maximal when the device is tuned to the charge neutrality region in the p-n junction. In addition, once the device is doped away from the charge neutral region no more interlayer photocurrent can be measured; at larger voltages away from neutrality the photoresponse is dominated by metallic-like photoconductivity (see our previous transport studies ref. 22 on this material system, as well as In Device 1 (Figure 2), we find that V G = -3.5 V exhibits the strongest reverse bias photocurrent, and as we tune the gate voltage within the charge neutrality window, we see qualitatively similar sideband behavior. To most clearly exhibit the photocurrent spectroscopy and sideband peaks, we concentrated on line-cuts of V G = -3.5 V where the photocurrent response is strongest. For other V G we also find the same periodic photocurrent sideband behavior, see Figure   S4.3. From this and additional characterization of Device 2 (Supplement Figure S4.1), we determine that tuning the gate voltage to V G = -3.5 V results in device response fully consistent with a gate-tunable p-n junction.
This value of the gate voltage where the strongest reverse bias photocurrent is achieved depends on specific details of the device geometry and extrinsic doping, and thus will vary from device to device. As a key distinction, similar behavior is observed for Device 2 shown in Figure  weaker. This occurs far away for the charge neutrality condition at gate voltages near 1 V and -1 V. This behavior is consistent with the behavior in Device 1, as described above.

S4.3 Data Intensive Exploration of Photocurrent Parameter Space
In this section, we explore the spatial photocurrent response of device 2 in greater detail, showing the MPDPM imaging datasets used to generate the data show in Fig. 3. Fig. S4.4a shows a set of 2,000 photocurrent images versus V G and V SD with the laser wavelength at 1250 nm (1.0 eV). This is dataset is spatially consistent, i.e. corrected for spatial drift, and from the imaging set we can see the large-scale behavior of the device. Zooming in on a subset of the images in Fig.   S4.4b and superimposing the outline of the heterostructure area of the sample, we see that the majority of the photocurrent originates outside the heterostructure, mainly from the semiconductor-metal interface at the device contacts, which will be discussed below. However, near V G = 0 V we observe distinct and spatially uniform photocurrent originating from the heterostructure area, consistent with our expectation of photocurrent from an interlayer exciton near charge neutrality.
To verify that the photocurrent from the heterostructure is due to the interlayer exciton we take a similar set of spatial photocurrent maps at a fixed value of V SD = -0.35 V while varying the laser wavelength, shown in Fig. S4.5a. Again, we identify the heterostructure region and consider the average photocurrent originating in that spatial area.
The heterostructure responsivity (photocurrent divided by laser power), is shown in Fig. 1.
We observed a bright stripe near V G = 0.25 V, corresponding to a resonance in photon energy peaked at 0.96 eV, very close to our expectation of a 1.0 eV interlayer exciton. Importantly, we find that the photocurrent diminishes as the device is tuned away from charge neutrality, indicating that this resonance corresponds to the interlayer exciton, which is consistent with the behavior described in detail in Section S4.2, above.  There are multiple spatially distributed sources of photocurrent in the imaging data sets, the majority of which come from the metal contacts. Fig. S4.6 visualizes the photocurrent at the three points in space corresponding to the principle sources of photocurrent, WSe 2 contact, the heterostructure and the MoSe 2 contact. Fig. S4.5a compares a photocurrent map and a reflection image and identifies the locations of the photocurrent as points. The heterostructure photocurrent in Fig. S4.6c has been well described in this work. The WSe 2 and MoSe 2 photocurrent in Fig.  S4.6b,d are both more active for positive values of V G and occur opposite to each other with respect to V SD . Photocurrent at the contacts results from the photo-thermoelectric effect at the metalsemiconductor interface, where the laser produces a local temperature gradient between two materials with significantly different Seebeck coefficients 28 .

S4.4. Analysis of Photocurrent Oscillations in the Current-Voltage Characteristics
In Fig. S4.7a we plot the average heterostructure photocurrent versus V SD and V G and observe a bright feature of positive photocurrent near V G = 0 as a function of increasing V SD . To proceed, we will look at the photoconductance, that is , which we plot in Fig. S4.7b. PC / SD We observe a bright feature corresponding to the main feature, but we also see a number of small oscillations in the conductance. To further visualize these oscillations, we take the second derivative , shown in Fig. S4.7c, where we see strong oscillations in the photocurrent. In Fig. 3, the approximate energy shift across the heterostructure is obtained by scaling the voltage V SD by two factors: and . The phenomenological factor is obtained from fitting the dark current to the diode equation, (shown in Fig. 1f), while is a = ( / -1) dimensionless parameter that characterizes the dipole length, as described in the main text. By analyzing the heterostructure photocurrent, we find that discretely spaced peaks in the photocurrent spectra emerge near charge neutrality. The interlayer photocurrent extracted from 2x10 3 photocurrent images (shown in Fig. S4.4a) occurs predominantly near V G = 0 V, increases as V SD increases in reverse bias (V SD < 0 V), and follows the diagonal trajectory (solid line in Fig. S4.8a). The heterostructure photocurrent parameter space which mainly consists of a bright feature that is diagonal with respect to V SD . b, Line cuts of the absolute value of photocurrent versus V SD , green points identify the maxima of the curves. c, Shows the V G value of the photocurrent maximum V G * versus V SD . The red line is the linear relation between V G * and V SD , which we see tracks the diagonal of the photocurrent in a.
By plotting the V G -dependent characteristics taken at several V SD values ( Fig. S4.8b), we extract the gate voltage values of the photocurrent peaks V G * , and plot V G * as a function of V SD ( Fig. S4.8c). Along the diagonal line in Fig. S4.7a, the interlayer electric field is such that the device maintains charge neutrality.

S4.4. Analysis of Photocurrent Oscillations at Different Temperatures
Having now converted the period of room temperature voltage oscillations into units of energy across the heterostructure, we can compare it to the low temperature photocurrent spectroscopy data. Fig. S4.9 compares the low temperature photocurrent 2D-FTs (from Fig. 2) and the room temperature imaging 2D-FT (from Fig. 3) with the phonon density of states (from Fig.   S3.1). We observe that in all cases there is a strong oscillation corresponding to the 30 meV phonon. Remarkably this oscillation is clearly observed even in the room temperature photoconductance oscillations, while less clear in the room temperature spectrum (Fig. S4.5b). The photocurrent oscillations observed in Fig. 3 and discussed above do not occur in the dark current of the system, confirming that they originate from the interlayer exciton. The photocurrent signal is measured using a lock-in amplifier to reject the large dark current. To confirm that the lock-in is functioning properly, and that the oscillations originate from the photocurrent we compare the photocurrent and photocurrent spectrum to the dark current. Fig.   S4.10a shows the second derivative of the photocurrent data from Fig. 3 and its Fourier transform; Fig. S4.10b shows the second derivative of the dark current over a similar range of and its Δ Fourier transform. Looking at the Fig. S4.10b we note that the dark current is dominated by high frequency noise and is not correlated with the 30 meV oscillation clearly observed in the photocurrent. Thus, we conclude that the oscillation originates from the photocurrent, and therefore the exciton, and not the dark current.

S5 Franck-Condon Transitions and Strong Exciton-Phonon Coupling
In this section, we discuss Franck-Condon transitions, the multiple phonon sidebands, as well as the conditions under which strong exciton-phonon coupling is manifested.

S5.1 Franck-Condon Transitions and Sidebands
We first review how multiple sidebands arise from Franck-Condon transitions and strong exciton-phonon coupling. We begin by analyzing an exciton transition from a ground state, , to an excited state, , where the excited state is strongly coupled with the atomic configuration of the system (captured by a phonon coordinate, ). For simplicity of presentation, we consider a single (optical) phonon mode with frequency , and its coupling to an exciton mode. The energy 0 surfaces for the ground and excited states can be described by 62 (S1) ( ) = where is the bare energy of the transition from ground to excited state in the absence of exciton-0 phonon coupling, is an elastic constant and is an effective mass characterizing the = 2 0 phonon mode. Here encodes the relative atomic displacement between the energy minima of 0 the electronic ground and excited state manifolds, which results from the exciton-phonon coupling.
For instance, when , there is no exciton-phonon coupling (Fig. 3a of the main text), yielding 0 = 0 an absorption that is characterized by a single broad absorption peak. In contrast, when , 0 ≠ 0 exciton-phonon coupling can lead to multiple side-band features in the absorption (Fig. 3b of the main text).
The multiple side-band features can be understood heuristically (see also explicit derivation below) by noting that the atomic configuration, characterized by the phonon coordinates, does not change in the short time over which an electronic transition takes place (the so-called "Franck-Condon principle") 63,64 . This means that optical transitions are "vertical" in the phonon coordinates. Since the energy surfaces are displaced by , optical transitions originating from the 0 minimum of the electronic ground state energy surface to the electronic excited state energy surface can leave the phonon mode in a range of excited states, thereby producing multiple sidebands. The strength of this effect is captured by the characteristic number of phonons involved in the transition, i.e., the average excitation number in the phonon state after the vertical transition.